How to Blur a Video Background in blender?
Yes, you can do it. There are two methods I can think of right off the bat:


Use a Difference Key! This will NOT work if your camera is moving, but can give you decent results that don't require masking by hand. If your footage is from a stationary camera that was mounted to a stable tripod, you could take the difference of the background with no actors, and then the background with actors. Just take the time to shoot the background alone, and then bring your actors in for another shot without moving the camera at all (you might want to just let the camera record continually for both shots so you don't have to touch it).


A word of warning though: if your actors appear in color that is
    similar to the background footage, the mask will have lots of nasty
    holes in it. So don't wear blue shirts if your background is blue,
    etc.

Also, the backgrounds in both shots have to stay EXACTLY the same; I would avoid anything that moves like trees in the wind etc because they will ruin the Difference Key. Cheap cameras also tend to auto adjust exposure / brightness, and this will also ruin your Difference Key if this happens. A good rule is to shoot in a bright area and do not get too close to the camera or block the light from entering it in any way. Also, watch out for shadows because they WILL show up with your actors, and cannot easily be separated (unless you mask them out by hand or try some layer filter curves to block darker color values).

Using a Difference Key on the two pieces of footage, you can apply a blur effect to your background while preserving the focus of your actors. In the image below, I have simulated a background shot, and a background shot with an actor, and used the Compositor to achieve fake DOF. You can use the same node set up for your video footage! You could further tweak the layers in Compositor to get more specific results, but I wanted to keep the example as simple as possible so you can build on it however you like. Notice on the "Difference Key" node I turned "Tolerance" to "0.009". You will need to play with "Tolerance" and "Falloff" to see how it affects your video footage for best results!



Use Tracking and Masking, and patience. If your footage has spots that work well for tracking (high contrast areas that are stable) you could use masks in Blender like the guy in this Youtube tutorial explains, showing how he did it on a still photo, but this same technique can be used with video, even video from a moving camera if you can track and animate the mask for the subject. The difficulty level to get good results varies a lot depending on what you want to do, and how complex your subject / scene is. Tracking and masking is awesomely powerful, and the scope of it is beyond what I can answer right now. I will say though, you will have to plan your shots well if you want the best tracking results, even placing markers or using props to make things easier when working with the footage in Blender. For Masking, it helps if you know how to use the Pen Tool / Bezier curve handles. Knowing those tools is essential for making good masks. Best wishes!